Newton's Law of Gravitation ratio

  • Thread starter brendan3eb
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  • #1
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Homework Statement


A mass M is split into two parts, m and M - m, which are then separated by a certain distance. What ratio m/M maximizes the magnitude of the gravitational force between the parts?

Homework Equations


F=Gm1m2/d^2

The Attempt at a Solution


I first just tried plugging in M - m and m in for m1 and m2 to see what I get.

F=(GMm-Gm)/d^2

I believe that I will have to take a derivative to find the maximum, but I am confused as to what I should take the derivative with respect to.
 

Answers and Replies

  • #2
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you should not need a derivative. look at extreme cases. if the ratio of M/m was 1000, what would the force be like? what if the ratio was smaller, say 2?
 
  • #3
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That is true, but I need to prove that is 1/2. The only way I can think of doing that is with a derivative. Although I do see the logic in what you are saying about taking the extreme cases.
 
  • #4
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try assigning integers to the variables M and d, since they do not change based upon different m values. also, recheck the GMm-Gm part of the equation. I got something else for if i plugged in the values like you did.
 
  • #5
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Starting with the regular equation for the force of gravity:

[tex]F=G\frac{m_0m_1}{d^2}[/tex]

Let [tex]m_0=M-m[/tex] and [tex]m_1=m[/tex]

Substitute into the first equation:

[tex]F=G\frac{(M-m)m}{d^2}=G\frac{Mm-m^2}{d^2}[/tex]

To maximize [tex]F[/tex], [tex]Mm-m^2[/tex] has to be maximized.

Take its derivative with respect to [tex]m[/tex] and set it to zero:

[tex]M-2m=0[/tex]

Rearrange:

[tex]M=2m[/tex]

Rearrange:

[tex]\frac{1}{2}=\frac{m}{M}[/tex]
 

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